I need to make a matrix/vector multiplication in Matlab of very large sizes: "A" is an 655360 by 5 real-valued matrix that are not necessarily sparse and "B" is a 655360 by 1 real-valued vector. My question is how to compute: B'*A efficiently.
I have notice a slight time improvement by computing A'*B instead, which gives a column vector. But still it is quite slow (I need to perform this operation several times in the program).
With a little bit search I found an interesting Matlab toolbox MTIMESX by James Tursa, which I hoped would improve the above matrix multiplication performance. After several trials, I can only have very marginal gains over the Matlab native matrix multiplication.
Any suggestions about how should I rewrite A'*B so that the operation is more efficient? Thanks.
Matlab's raison d'etre is doing matrix computations. I would be fairly surprised if you could significantly outperform its built-in matrix multiplication with hand-crafted tools. First of all, you should make sure your multiplication can actually be performed significantly faster. You could do this by implementing a similar multiplication in C++ with Eigen.
I have had good results with matlab matrix multiplication using the GPU
In order to avoid the transpose operation, you could try:
sum(bsxfun(#times, A, B), 2)
But I would be astonished it was faster than the direct version. See #thiton's answer.
Also look at http://www.mathworks.co.uk/company/newsletters/news_notes/june07/patterns.html to see why the column-vector-based version is faster than the row-vector-based version.
Matlab is built using fairly optimized libraries (BLAS, etc.), so you can't easily improve upon it from within Matlab. Where you can improve is to get a better BLAS, such as one optimized for your processor - this will enable better use of the caches by getting appropriately sized blocks of data from main memory. Take a look into creating your own compiled versions of ATLAS, ACML, MKL, and Goto BLAS.
I wouldn't try to solve this one particular multiplication unless it's really killing you. Changing up the BLAS is likely to lead to a happier solution, especially if you're not currently making use of multicore processors.
Your #1 option, if this is your bottleneck, is to re-examine your algorithm. See this question Optimizing MATLAB code for a great example of how choosing a different algorithm reduced runtime by three orders of magnitude.
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As a hobby project I'm taking a crack at finding really large prime numbers. The primality tests for this contain modular exponentiation calculations, i.e. a^e mod n. Let's call this the modpow operation to keep the explanation simple. I am wanting to speed up this particular calculation.
Currently I am using GMP's mpz_pown function, but, it is kind of slow. The reason I think it's too slow, is because a function call to GMP's modpow is slower than a full-blown primality test of the software called PFGW for the same large number. (So to be clear, this is just the GMP's modpow part, not my whole custom primality testing routine I am comparing). PFGW is considered the fastest in it's field and for my use case it uses a Brillhart-Lehmer-Selfridge primality test - which also uses the modpow procedure - so it's not because of mathematical cleverness that PFGW is faster in that aspect (please correct me if I'm wrong here). It looks like the bottleneck in GMP is the modpow operation. An example runtime for numbers which have a little over 20,000 digits: GMP's modpow operation takes about 45 seconds and PFGW finishes the whole primality test (involving a modpow) in 9 seconds flat. The difference gets even more impressive with even bigger numbers. GMP uses FFT multiplication and Montgomery reduction for this test comparison, see comments on this post below.
I did some research. So far I understand that the modpow algorithm uses exponentiation by squaring, integer multiplication and modulo reduction - these all sound very familiar to me. Several helper methods could improve the running time of integer multiplication:
Montgomery multiplication
FFT multiplication
To improve the running time of the exponentiation by squaring part, one may use a signed digit representation to reduce the number of multiplications (i.e. bits are represented as 0, 1 or -1, and the bit string is represented in such a way so that it contains many more zeros than in it's original base-2 representation - this reduces the running time of exponentiation by squaring).
For optimizing the modulo part of the operation, I know of these methods:
Montgomery reduction
So here is the 150,000 dollar question: is there a software library available to do a modpow operation efficiently given a very large base, exponent and modulus? (I'm aiming for several millions of digits). If you would like to suggest an option, please try to explain the inner workings of the algorithm for the case with millions of digits for the base, modulus and exponents, as some libraries use different algorithms based on the number of digits. Basically I am looking for a library which supports the techniques mentioned above (or possibly more clever techniques) and it should perform well while running the algorithm (well, better than GMP at least). So far I've searched, found and tried GMP and PFGW, but didn't find these satisfying (PFGW is fast, but I'm just interested in the modpow operation and there is no direct programming interface to that). I'm hoping that maybe an expert in the field can suggest a library with these capabilities, as there seem to be very few that are able to handle these requirements.
Edit: made the question more concise, as it was marked too broad.
First off, re. the Answer 1 writer's comment "I do not use GMP but I suspect when they wrote they use FFT
they really mean the NTT" -- no, when GMP says "FFT' it means a floating-point FFT. IIRC they also have some NTT-based routines, but for bignum mul those are uncompetitive with FFT.
The reason a well-tuned FFT-mul beats any NTT is that the slight loss of per-word precision due to roundoff error accumulation is more than made up for by the vastly superior floating-point capabilities of modern CPU offerings, especially when one considers high-performance implementations which make use of the vector-math capabilities of CPUs such as the x86_64 family, the current iterations of which - Intel Haswell, Broadwell and Skylake - have massive vector floating-point capability. (I don't cite AMD in this regard because their AVX offerings have lagged far behind Intel's; their high-water mark was circa 2002 and since then Intel has been beating the pants off them in progressively-worse fashion each year.) The reason GMP disappoints in this area is that GMP's FFT is, relatively speaking, crap. I have great respect for the GMP coders overall, but FFT timings are FFT timings, you don't get points for effort or e.g. having a really good bignum add. Here is a paper detailing a raft of GMP FFT-mul improvements:
Pierrick Gaudry, Alex Kruppa, Paul Zimmerman: "A GMP-based Implementation of Schönhage-Strassen's Large Integer Multiplication Algorithm" [http://www.loria.fr/~gaudry/publis/issac07.pdf]
This is from 2007, but AFAIK the performance gap noted in the snippet below has not been narrowed; if anything it has widened. The paper is excellent for detailing various mathematical and algorithmic improvements which can be deployed, but let's cut to the money quote:
"A program that implements a complex floating-point FFT for integer multiplication is George Woltman’s Prime95. It is written mainly for testing large Mersenne numbers 2^p − 1 for primality in the in the Great Internet Mersenne Prime Search [24]. It uses a DWT for multiplication mod a*2^n ± c, with a and c not too large, see [17]. We compared multiplication modulo 2^2wn − 1 in Prime95 version 24.14.2 with multiplication of n-word integers using our SSA implementation on a Pentium 4 at 3.2 GHz, and on an Opteron 250 at 2.4 GHz, see Figure 4. It is plain that Prime95 beats our im- plementation by a wide margin, in fact usually by more than a factor of 10 on a Pentium 4, and by a factor between 2.5 and 3 on the Opteron."
The next few paragraphs are a raft of face-saving spin. (And again, I am personally acquainted with 2 of the 3 authors, and they are all top guys in the field of computational number theory.)
Note that the aforementioned George Woltman, whose Prime95 code has discovered all of the world-record primes since shortly after its debut 20 years ago, has made his core bignum routines available in a general API-ized form called the GWNUM library. You mentioned how much faster PFGW is than GMP for FFT-mul - that's because PFGW uses GWNUM for the core 'heavy lifting' arithmetic, that's where the 'GW' in PFGW comes from.
My own FFT implementation, which has generic-C build support but like George's uses reams of x86 vector-math assembler for high performance on that CPU family, is roughly 60-70% slower than George's on current Intel processor families. I believe that makes it the world's 2nd-fastest bignum-mul code on x86. By way of example, my code is currently running a primality test on a number with roughly 2^29 bits using a 30-Mdouble-length FFT (30*2^20 doubles); thus a little more than 17 bits per input word. Using all four of my 3.3 GHz Haswell 4670 quad's cores it takes ~90 ms per modmul.
BTW, many (if not most) of the world's top bignum-math coders hang out at mersenneforum.org, I encourage you to check it out and ask your questions to the broader (at least in this particular area) expert audience there. I appear under the same handle there as here; George Woltman appears as "Prime95', PFGW's Mark Rodenkirch goes as "rogue".
I do not use GMP at all so handle this with that in mind.
I rather use NTT instead of FFT for multiplication
it removes the rounding errors and in comparison to mine FFT implementations optimized to the same point is faster
C++ NTT implementation
C++ NTT sqr(mul) implementation
as I mentioned I do not use GMP but I suspect when they wrote they use FFT they really mean the NTT (finite field Fourier transform)
The speed difference of your test and the GMP primality test can be caused by modpow call.
if there are too much calls to it then it causes heap/stack trashing which slows things down considerably. Especially for bignums. Try to avoid heap trashing so eliminate as much data from operands and return calls as possible for frequently called functions. Also sometimes helps to eliminate the bottleneck call by directly copy the function source code into your code instead of the call (or use of macros instead) with use of local variables only.
I think GMP published their source code so finding their implementation of modpow should not be too hard. You just have to use it correctly
just to be clear
you are using big numbers like 20000+ decadic digits which means ~8.4 KBytes per number. Any return or non pointer operand means to copy that amount of data to/from heap stack. This not only takes time but also usually invalidates the CPU's CACHE which also kills performance.
Now multiply this by the algorithm iterations number and you get the Idea. Had similar problems while tweaking many of mine bignum functions and the speedup is often more than 10000% (100 times) even if there is no change in the algorithm used. Just limiting/eliminating heap/stack trashing.
So I do not think you need better modpow implementation just better usage of it but of coarse I may be wrong but without the code you are using is hard to deduce more.
I have a problem that requires me to do eigendecomposition and matrix multiplication of many (~4k) small (~3x3) square Hermitian matrices. In particular, I need each work item to perform eigendecomposition of one such matrix, and then perform two matrix multiplications. Thus, the work that each thread has to do is rather minimal, and the full job should be highly parallelizable.
Unfortunately, it seems all the available OpenCL LAPACKs are for delegating operations on large matrices to the GPU rather than for doing smaller linear algebra operations inside an OpenCL kernel. As I'd rather not implement matrix multiplcation and eigendecomposition for arbitrarily sized matrices in OpenCL myself, I was hoping someone here might know of a suitable library for the job?
I'm aware that OpenCL might be getting built-in matrix operations at some point since the matrix type is reserved, but that is not really of much use right now. There is a similar question here from 2011, but it pretty much just says to roll your own, so I'm hoping the situation has improved since then.
In general, my experience with libraries like LAPACK, fftw, cuFFT, etc. is that when you want to do many really small problems like this, you are better off writing your own for performance. Those libraries are usually written for generality, so you can often beat their performance for specific small problems, especially if you can use unique properties of your particular problem.
I realize you don't want to hear "roll your own" but for this type of problem it is really the best thing to do IMO. You might find a library to do this, but considering the code that you really want (for performance) will not generalize, I doubt it exists. You'll be looking specifically for code to find the eigenvalues of 3x3 matrices. That's less of a library and more of a random code snippet with a suitable license that you can manipulate to take advantage of your specific problem.
In this specific case, you can find the eigenvalues of a 3x3 matrix with the textbook method using the characteristic polynomial. Remember that there is a relatively simple closed form solution for cubic equations: http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots.
While I think it is very likely that this approach would be much faster than iterative methods, it would be wise to verify that if performance is an issue.
libsvm and liblinear are both software libraries that implement Support Vector Machines. What's the difference? And how do the differences make liblinear faster than libsvm?
In practice the complexity of the SMO algorithm (that works both for kernel and linear SVM) as implemented in libsvm is O(n^2) or O(n^3) whereas liblinear is O(n) but does not support kernel SVMs. n is the number of samples in the training dataset.
Hence for medium to large scale forget about kernels and use liblinear (or maybe have a look at approximate kernel SVM solvers such as LaSVM).
Edit: in practice libsvm becomes painfully slow at 10k samples.
SVM is support vector machine, which is basically a linear classifier, but using many kernel transforms to turn a non-linear problem into a linear problem beforehand.
From the link above, it seems like liblinear is very much the same thing, without those kernel transforms. So, as they say, in cases where the kernel transforms are not needed (they mention document classification), it will be faster.
From : http://www.csie.ntu.edu.tw/~cjlin/papers/liblinear.pdf
It supports L2-regularized logistic regression (LR), L2-loss and L1-loss linear support vector machines (SVMs) (Boser et al., 1992). It inherits many features of the popular SVM library LIBSVM
And you might also see some useful information here from one of the creators: http://agbs.kyb.tuebingen.mpg.de/km/bb/showthread.php?tid=710
The main idea, I would say, is that liblinear is optimized to deal with linear classification (i.e. no kernels necessary), whereas linear classification is only one of the many capabilities of libsvm, so logically it may not match up to liblinear in terms of classification accuracy. Obviously, I'm making some broad generalizations here, and the exact details on the differences are probably covered in the paper I linked above as well as with the corresponding user's guide to libsvm from the libsvm website.
This should be very simple but I could not find an exhaustive answer:
I need to perform A+B = C with matrices, where A and B are two matrices of unknown size (they could be 2x2 or 20.000x20.000 as greatest value)
Should I use CUBLAS with Sgemm function to calculate?
I need the maximum speed achievable so I thought of CUBLAS library which should be well-optimized
For any sort of technical computing, you should always use optimized libraries when available. Existing libraries, used by hundreds of other people, are going to be better tested and better optimized than anything you do yourself, and the time you don't spend writing (and debugging, and optimizing) that function yourself can be better spent working on the actual high-level problem you want to solve instead of re-discovering things other people have already implemented. This is just basic specialization of labour stuff; focus on the compute problem you want to solve, and let people who spend their days professionally writing GPGPU matrix routines do that for you.
Only when you are sure that existing libraries don't do what you need -- maybe they solve too general a problem, or make certain assumptions that don't hold in your case -- should you roll your own.
I agree with the others that in this particular case, the operation is pretty straightforward and it's feasible to DIY; but if you're going to be doing anything else with those matricies once you're done adding them, you'd be best off using optimized BLAS routines for whatever platform you're on.
What you want to do would be trivial to implement in CUDA and will be bandwidth limited.
And since CUBLAS5.0, cublasgeam can be used for that. It computes the weighted sum of 2 optionally transposed matrices.
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What is the best matrix multiplication algorithm? What means 'the best'for me? It means the fastest and ready for todays machines.
Please give links to pseudocode if you can.
BLAS is the best ready-to-use efficient matrix multiplication library. There are many different implementation. Here is a benchmark I made for some implementations on a MacBook Pro with dual-core Intel Core 2 Duo 2.66 GHz :
gotoBLAS2 (open-source) : https://www.tacc.utexas.edu/research-development/tacc-software/gotoblas2
ATLAS (open-source) : http://math-atlas.sourceforge.net/
Accelerate.framework (Apple) : http://developer.apple.com/performance/accelerateframework.html
a non-optimized, but portable, implementation that I called 'vanilla' (from the GSL)
There are also other commercial implementations that I didn't test here :
MKL (Intel) : http://software.intel.com/en-us/articles/intel-mkl/
ACML (AMD) : http://developer.amd.com/cpu/Libraries/acml/Pages/default.aspx
The best matrix multiplication algorithm is the one that someone with detailed architectural knowledge has already hand-tuned for your target platform.
There are lots of good libraries that supply tuned matrix-multiply implementations. Use one of them.
There are probably better ones but these are the ones I've head of (better than the standard cubic complexity algorithm).
Strassen's - O(N^2.8)
Coppersmith Winograd - O(N^2.376)
Why pseudocode? Why implement it yourself? If speed is your concern, there are highly optimized algorithms available that include optimizations for specific instruction sets (e.g. SIMD), implementing those all by yourself offers no real benefit (apart from maybe learning),
Take a look at different BLAS implementations, like:
http://www.netlib.org/blas/
http://math-atlas.sourceforge.net/
Here is algorithms course of MIT and the matrix multiplication lecture
http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/video-lectures/lecture-19-shortest-paths-iii-all-pairs-shortest-paths-matrix-multiplication-floyd-warshall-johnson/
matrix multiplication - O(n^3)
Strassen’s algorithm - O(n^2.8) http://en.wikipedia.org/wiki/Strassen_algorithm
Coppersmith–Winograd - O(n^2.376) http://en.wikipedia.org/wiki/Coppersmith%E2%80%93Winograd_algorithm
Depends on the size of the matrix, and whether it's sparse or not.
For small-to-medium-sized dense matrices, I believe that some variation on the "naive" O(N^3) algorithm is a win, if you pay attention to cache-coherence and use the platform's vector instructions.
Data arrangement is important -- for cases where your standard matrix layout is cache-unfriendly (e.g., column-major * row-major), you should try binary decomposition of your matrix multiplication -- even if you don't use Strassen's or other "fast" algorithms, this order of operations can yield a "cache-oblivious" algorithm that automatically makes good use of every level of cache. If you have the luxury to rearrange your matrices, you might try combining this with a bit-interleaved (or "Z-order") ordering of data elements.
Finally, remember: premature optimization is the root of all evil. And when it's not premature any more, always profile & benchmark before, during, and after optimizing....
There is no "best algorithm" for all matrices on all modern CPUs.
You will need to do some research into the many methods available, and then find a best-fit solution to the particular problems you are calculating on the particular hardware you are dealing with.
For example, the "fastest" way on your hardware platform may be to use a "slow" algorithm but ask your GPU to apply it to 256 matrices in parallel. Or using a "fast" general-purpose (mxn) algorithm may produce much slower results than using an optimised 3x3 matrix multiply. If you really want it to be fast then you may want to consider getting down to the bare metal to make sure you make best use of specific CPU features like SIMD instructions, branch prediction and cache coherence, at the expense of portability.
There is an algorithm call the Cannon's algorithm a distributed matrix multiplication algorithm. More here